Economics Department of the University of Pennsylvania
Institute of Social and Economic Research -- Osaka University
Decision Making and the Temporal Resolution of Uncertainty
Author(s): Larry G. Epstein
Reviewed work(s):
Source: International Economic Review, Vol. 21, No. 2 (Jun., 1980), pp. 269-283
Published by: Blackwell Publishing for the Economics Department of the University of Pennsylvania and
Institute of Social and Economic Research -- Osaka University
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INTERNATIONAL ECONOMIC REVIEW
Vol. 21, No. 2, June, 1980
DECISION MAKING AND THE TEMPORAL
RESOLUTION OF UNCERTAINTY*
BY
LARRY G. EPSTEIN1
1.
INTRODUCTION
There exist in the literature many two-period analyses of behavior under
uncertainty. (See, for example, Sandmo [1970] and [1971], Rothschild and
Stiglitz [1971], Turnovsky [1973] and Epstein [1975, 1978].) These works typically assume that a decision must be made in period 1 subject to uncertaintyabout
the environment that will prevail in period 2. At the start of period 2 the true
state of the environment becomes known and perhaps some further decisions are
made. The effects on the period 1 decision of the prior uncertaintyin expectations
are closely examined.
Clearly the above framework is inadequate for modelling the more general
situation where n >1 decisions are made sequentially and subject to improving
information about the eventual state of the world. In this general framework
the influence on decisions of the way in which uncertainty is resolved through
time represents an interesting area of investigation. This paper undertakes such
an investigation in the context of several specific decision problems.
For simplicity (see footnote 4), we adopt a minimal extension of the two-period
model that makes possible the analysis of the temporal resolution of uncertainty,
namely, a three-period model. Thus the decision-making framework considered
in this paper may be described as follows: an expected utility maximizing agent
faces a three-period planning horizon. He makes decisions in each period.
Period 3 decisions are made after all uncertainty has been resolved. In period 1
the decision is made subject to prior expectations about the state of the world that
will prevail in period 3. The uncertain future environment is represented by a
random variable Z. Before the start of period 2 new information about the
ultimate value of Z becomes available. The information is forthcoming through
the observation of another random variable Y which in general is correlated with
Z. The agent is a Bayesian decision maker and revises his prior probability
distribution about Z after observing Y. The amount of additional information
about Z provided by Y is a parameter in the model.
The objective of the paper is to compare the decisions in period I in two choice
problems that differ only with respect to the amount of information provided by
Y about Z. Note that in both problems the agent faces the identical prior uncertainty about Z. The difference in the two problems is only in the amount of
* ManuscriptreceivedJune 1, 1978, revisedFebruary21, 1979.
The helpfulcommentsof two refereesare gratefullyacknowledged.
1
269
270
LARRY G. EPSTEIN
information he can expect to gain about Z before his period 2 decision is made.
(Two extreme cases are (a) no additional information, and (b) perfect informationl.
In the latter case the period 2 decision is made under certainty and we have a
two-period framework.) Thus it seems justifiable to describe the analysis below
as an investigation of the effects on decision making of the temporal resolution of
uncertainty. We will adopt the following terminology: if Y' containis more (less)
information about Z than does Y, uncertainty is considered to be resolved earlier
(later) in the decision problem characterized by Y' than in that characterized
by y2
The notion of greater information adopted in this paper is that due to Blackwell
[1951, 1953] and Marschak and Miyasawa [1968], and employed by Kihlstrom
[1974] and Grossman, Kihlstrom and Mirman [1977]. Section 2 describes the
notion briefly, relying heavily on Marschak and Miyasawa (MM). Of particular
importance is Theorem 1 which describes the way in which the behavioral implications of greater future information may be determined. (It plays an
analogous role here to that played by the Rothschild and Stiglitz result [1971,
p. 67] in the comparative statics analysis of greater uncertainty.)
Note the difference between our study and the two studies above which employ
the same definition of greater information. In the latter, the amount of
information is endogenous ainddetermined by the agent subject to the costs of acquisition. Here, the amount of information is exogenous to the agent and is
forthcoming automatically with the passage of time.3 Partly as a result of our
simpler framework we are able, unlike the other studies, to derive comparative
statics propositions which are perfectly general in the sense that they do not depend
on a particular specification of the joint probability distribution of Y and Z.
On intuitive grounds one would expect the prospect of greaterfuture information
to increase the incentive to maintain some flexibility in period 2 to take advantage
of the information. This incentive would discourage the adoption in period 1
of decisions that would limit the agent's options in period 2. Such a behavioral
response to greater information is readily established in a model where a totally
irreversibledecision, i.e., one that fixes the decision in period 2 as well, is in the
agent's period 1 choice set. It is shown in Section 3 that the prospect of greater
future information discourages the adoption of an irreversible decision. The
result is trivial to derive and the model is not of much interest in itself.
It is included in order to provide some perspective regarding the other, nontrivial
results in the paper that concern less extreme irreversibilities. For example, in
2 We emphasizethat by uncertaintyresolvingearliergiven Y rather than Y' we do not mean
that the decision makerlearns Y earlierin the sequenceof decision making.
3 More generallywe might assume an exogenous cost of informationacquisitionand investigate the consequencesof variationsin that cost. Our results, suitably modified, will iemain
valid if: (i) informationcosts enter the individual'sobjectivefunction in an additively separable
fashion, and (ii) the demandfor informationis a downwardsloping function of the acquisition
cost. This is the case for a reasonableextension of the models in Sections 5 and 7. (See also
Kihlstrom[1976].)
TEMPORAL RESOLUTION OF UNCERTAINTY
271
Sections 4 and 5 the intuitively expected results are derived in more interesting
models but only for a more restrictive class of (intertemporally additive) utility
functions than conisideredin Section 3. A simple consumption-savings model is
considered in Section 6. Consumption of period 1 wealth is an irreversible decision in that it limits futuLreconsumption-savings options. However, optimal
first period consumptioin does not unambiguously fall in response to an earlier
resolution of uncertainty. The corresponding response in the model of the firm
analyzed in Section 7 is similarly ambiguous. In both cases we are able to
determine the parameters Liponwhich the qualitative responses depend.
Also undertaken in the context of the last two models is a comparison between
thlequalitative effects of (i) anl earlier resoluLtionof a given prior uncertainty, and
(ii) a reduction in the prior variability of expectations for a given structure of
resoltutioniover time. Clearly the expected utility of all risk averters is unambiguously raised by either chanigein expectations. One might consequently be led to
guess that (i) and (ii) would have similar qualitative impacts on behavior. In
fact this is false. We show that there does not exist a general relationship between
the qualitative effects of (i) and (ii) on period 1 decisions. Related is the observation that the behavioral consequences of reduced prior variability depend on
the way in which the uncertainty is resolved over time.
A few words are in order concerning the literatureon1the structureof preferences
in problems of dynamic choice. Mossin [1969], Dr&zeand Modigliani [1972]
and Spence and Zeckhauser [1972] discuss the preference for random income
streams induced from primitive preference for consumption streams defined by a
von Neumann-Morgenstern Litilityindex. They observe that such induced preference is not consistent with the von Neumann-Morgenstern axioms and in
particular that the timing of the resoluLtionof uncertainty concerning income can
be important. Kreps and Porteus [1978a] generalize the von NeumannMorgenstern axioms to permit the temporal resolution of uncertainty to matter.
In [1978b] the authors compare the generalized preference structure with induced
preferencefor random income streams. Kreps [1978] characterizesa "preference
for flexibility" which is closely related to the preference structure adopted in
this paper. Finally, Selden [1978] provides an alternative axiomatization of
preference for dynamic clhoice problems. This paper investigates the behavioral
consequenicesof the manner in which tincertainty resolves given that preference
for random variables is induced from a primitive von Neumann-Morgenstern
utility index. A comparable analysis in the context of one of the alternative
preference structures mentioned above would be an interesting subject for future
research.
LARRY G. EPSTEIN
272
2.
THE GENERAL DECISION PROBLEM
This paper considers some special instances of the following general decision
problem:4
(1)
max Ey max {EzlyU(xl, x2, Z) IX2 E C2(X1)}X
xjeCj
X2
xl and x2 are real scalars that represent period 1 and period 2 decisioni variables
respectively. C, and C2(x,) are convex subsets of the non-negative real line with
nonempty interiors.5 U is a von Neumann-Morgenstern utility index that is
concave and twice continuously differentiable ill (x,, x2). Z is a random variable
(r.v.) reflecting the agent's subjective tincertainty about his future economic
environment. The true value of Z becomes known at the end of period 2.
Before x2 is chosen, however, the agent gains some information about Z by
observing a r.v. Y whiclh in general is correlated with Z. The individual is a
Bayesian decision maker. His prior probability distribution for Z, held at the
start of period 1, is revisedaccording to Bayes' Rule after observing Y. xl is chosen
before observing Y. However, the prospect of future information about Z is,
of course, taken into account by the agent when choosing xl. C2(x,) represents
a constraint set, in general depending on xl, which faces the agent in choosing x2.
The objective of the paper is to investigate how the clhoiceof xl is influenced by
varying amounts of information provided by Yabout Z. We adopt the nlotion of
"greater information" due to Blackwell [1951] and [1953] and discussed by MM
[1968]. We follow the MM discussion. Adopting their notation we may briefly
describe the results of their analysis which are important below. (For further
details the reader should refer directly to their work.)
Y and Z are assumed to be discrete r.v.'s with possible realizations (y,,
Y.)
and (zz,..., z,,) respectively. The set {Yi,*,. y,,} is called the set of possible
messages corresponding to Y. The corresponding probability vectors are qT=
and -T=(r/1,...,
The likelihood
,,), ii=Pr (Z=zi).6
(q15... qn), qi=Pr(Y=yi)
matrix is denioted A=(1'ij), where Xij=Pr(Y=yjJZ=zi) and the posterior probYis often called
ability distribuLtionis denoted H=(7rij),
rij=Pr(Z=ziJY=yj).
an experiment. If Y' is another experiment, (y'1,..., y,), q', A' an I1 are defined
I The general multi-periodproblem involves a sequence of decisions
x, and experiments
representedby Y,, t--l, 2,..., T, with x, chosen after the realization of Y,-1 but before the
realizationof Y,. Y, generally provides informationabout each Yt+k, k>O. The efTectsof
a change in the informationstructuremay be determinedby extensions of the argumentsin
this paper. As an example consider the consequencesof varying the informativenessof Y1
with respectto Y2alone keepingunchangedall other conditional distributions Y,/(Y,-i,..., YI),
t>2. The effects of such a change may be analyzed in a three period frameworkwhere U is
a utility function derivedfrom the underlyingmulti-periodutilityfunction by standarddynamic
programmingarguments.
I The single exceptionis in Section 3 wherexl and x2 may assumediscretevalues only.
6 All vectorsare column vectors unless transposedby a superscriptT. Note also that partial
derivativesare denoted throughoutby appropriatesubscriptedvariables.
273
TEMPORAL RESOLUTION OF UNCERTAINTY
in an obvious fashion. Z and its prior probability distribution r are fixed and
the effects of a clhange in experiment from Y to Y' are considered. (Clearly,
therefore, qT = -TA, q'l' =.TA', and Hq= H7'q'=er.)
Y is said to be more informative thanl Y' if every user of information about Z
is at least as well off observing Y before making a decision as he would be if he
based his decision on an observation of Y'. This definition may be made more
I
precise in the context of decision problems of the type (1). Let Sm'-1
(2)
max {Z
EiU(x 1,
J(x1,,)
X2,
Z)
IX2 e
C2(X
1
i
X2
Denote by 7rjanid7r the j-th columns of H and I7'respectively. Then Y is more
informative then Y' if and only if:
(D.1)
_q'J(x,, 7?) < _qjJ(x1, 7rj), for all x,, U and C2 for which the maxi
J
imLIm in (2) exists.7 (Two extreme experiments are of interest. Y provides no
information about Z if Y and Z are stochastically independent. Y provides
perfect information if Z conditional on Y is certain.)
Several alternative characterizations of D.1 are described by MM.8 One
(due initially to Blackwell) which is important for the purpose of comparative
statics anialyses is the following (Theorem 12.1): Y is more informative than Y'
if and only if
(D.2)
Z
j=1
qjp(Tj) ?
Z
j=1
q,p(7r),
for any convex function p on S"'-'.
The application of this characterization in this paper stems from the following
theorem. It is assumed that the maxima in (1) exist and are unique and for simplicity, that the optimal x, lies in the interior of C,. The function J(x1, 4) defined
by (2) is assumed to be concave and differentiablc with respect to x,. (That will
be the case if U and C2 are "well-behaved" as we assume in all the examples considered below. For example, J(x,, 4) is (strictly) concave in x, if U(x,, x2, z)
is (strictly) concave in x,, x2 and if C2(x,) satisfies the following condition:
X2 e C2(XI),
t2
e C2(tl)
=
>
/X2
+ (1
-
A)t2
E C20AX1
+ (1
-
)tl),
0 <
A < 1.
I Thc relationshipof this notion of informationto entropy is clarified by Marschak[1974]
and Maischakand Miyasawa[1968). Specificparameterizationsof the amount of information,
consistent with the general definition,are adopted by Kihlstrom[1974),Grossman, Kihlstrom
and Mirman[1977), Bradfordand Kclejian[1977)and Murota[1976). Bradfordand Kelejian
assumc essentiallythat Y:-Z-i-b where 6, the observationor predictionerror, is normallyand
indepcndentlydistributedwith zeIo mcan. The amounitof informationprovided by Y varies
inversely with the variance of o. Murota assumes ,z--n-3, (ZI, z2, Z3)=--(YI, Y2, y3) and
1
for ij and -s for i *j. The amount of informationvariesinversely
Pr(Y- yil ==--j)--1-2e
witlh S.
8 Green and Stokey [1978) relate these characterizationsof "more informative"to another
involving the set S of partitionsof the set of possible outcomes of a given experiment. S is
naturallyorderedby the criterionof refinement.
274
LARRY G. EPSTEIN
This condition is satisfied if C2(x1)={x2Jf(x1,
X2)20},
f concave.)
THEOREM 1. Let x4 an1dx** be the solutions of (1) given Yand Y' respectively,
where Y is nore infornative than Y'. Let J(xl, C), E- S"I-', be defined by (2).
If J,(x1, 4) is neitheer
If J1(x*, 4) is concave (convex) in ,, then x*<(>)x**.
convex nor concave, then the sign of x1-4x1* is ambiguous in thefollowing sense:
there exist r.v.'s Z, Y, Y' and Y" stch that Y' and Y" are each less informative
than Y and such that the optimal xl for Y' exceedlsxl and that foi Y" is less
than x.
PROOF.
By assumption x* and x** are the unique solutions to
ZqjJ,(x4, 7rj)= 0 and Yq>Jx,(x4**, t)=
0
respectively. Suppose J,1(x, 4) is convex in 4. Since Yis more informative than
Y', 0=YqjJx1(x, 7rj)? YqVJx(x*, 7r'). Therefore x*<x? . Similarly, J,1(x*
The
4) convex in 4 and it follows thlat x;**>x.
4) concave in =>>-J,1(x,
final assertion in the theoiem is analogous to a similar statement in Rothschild
and Stiglitz [1971, p. 67] concerning the comparative static effects of increased
uncertainty. The proof here is also analogous to the proof of the RothschildStiglitz assertion which is based on the analysis in Rothschild and Stiglitz [1970,
p. 240], and would require a straightforwardextension of Theorem 12.1 of MM.
Several comments are in order concerning the framework described in (1) and
concerning Theorem 1 before proceeding to applications of the latter. First
note that the strict inequalities x*<(>)x1* may be established if ZqjJ,1(x,
7rj)< (>)Y qJ,1(xi, 7i). An extension of the MM analysis shows that the latter
will be the case if the posterior probability distributions it and 7t' are not equal
and if J,1(x, 4) is strictly concave (convex) in 4, or indeed (see Sections 4 and 5)
only suitably nonlinear in 4.
Second, if the constraint defined by C1 is binding given either Y ol Y', tlle
qualitative relationship described in the theorem between x1 and xl* remains
valid. (For example, suppose that the constraint x1 > 0 is binding given Y.
Then YqjJ,1(0, 7rj)<O. Therefore J,1(0, 4) concave (convex)=:>q>Jx1(0, 7r)
? (< )0=*x* ? (<)x= -0.) For simplicity we shall assume interior solutions
for x1 in the problems below.
The decision problem (1) is sufficiently general to include models where a
third decision X3 must be made after Z is observed and all uncertainty is removed.
The function U in (1) need only be interpretedas a derived utility function, derived
from a more basic underlying function after optimizing with respect to X3. Thus
(1) does representthe three period decision problem referredto in the introduction.
Finally, some of the results discussed by MM have been proven for continuous
random variables by Blackwell [1951] and [1953]. However, it does not appear
that such an extension of the important Theorem 12.1 has been established.
Therefore, all r.v.'s in the sequel are assumed to be discrete.
TEMPORAL RESOLUTION OF UNCERTAINTY
AN IRREVERSIBLE
3.
275
DECISION9
Consider problem (1) modified so that xl and x2 can each take on the values
0 and 1 only. Choosing x1 =1 is an irreversible decision in the sense that
C2(1)= {1}. The alternative decision x1 =0 is such that C2(0)= {o, 1}.
Because futuLeiniformationcan be of value only if x1 =0, the prospect of more
iiiformation in the future would tenldto discourage the adoption of an irreversible
diescision in period 1. More tormally, adopt the notation of Section 2 with Y
being more informative than Y' and x4 and xl* the correspondinigoptimal decisions. We show that xl* = 0 = > x* = 0.
By D.1,
Suppose that xl*=0 and henice that _qVJ(O it)>
_q)J(1, 7).
Also, Y_qVJ(l 7,)=-_Yq, 7jU(l 1, Zi)= 7ijU(1,
EqjJ(O, 7j)> Y qJ(O, 7,).
J
1, zi)=qjJ(1,
7j). It follows that ZqjJ(O, rj)>ZqjJ(l, 7rj).
?
4.
HIGHWAYS
AND FARMS
Conisidera planner who solves the following problem:
(3)
max u(x1, 1 - x1) +
1?x1?O
qj max
Enijv(x2,
1 - x2, z).
X2<XI
There is one ullit of farm land available initially. The planner has the option
of pavinlg over as much of that land, in the form of highways, as he wishes in
period 1. xl representsthe amount of farm land left intact and 1-xl the amount
turned into highway. u(f, h) is a cardinal index of social utility derived in period 1
from farms and highways of sizes f and hIrespectively. The planner has a two
period horizon. At the start of period 2 he may convert more farms into
highways. x2 represents the amnountof farm land left intact for period 2. The
constraint x2<x1 corresponds to the fact that the transformation of farms into
highway is irreversible. This irreversibility poses a problem to the planner in
period I because there is uncertainty about the utility that will be derived in
period 2 from anyf and hi. Thus the second period utility index v(f, 11,zi) depends
on the realization zi of a random variable Z. The planner entertains a prior
probability distribution for Z when xl is chosen. Additional information about
Z is forthcoming, however, through an observation of a r.v. Y before x2 is chosen.
qj and rij are as defined in Section 2. u and v are assumed to be increasing and
strictly concave in (f, hi).
Intuitively one would expect that the more information that will be available
about Z before the period 2 decision must be made, the less farm land will the
planner decide to pave in period 1. This is in fact the case as we now show.
Define
I
This section applies D.1 rattherthan D.2 or Theorem 1. See footnote 5.
LARRY G. EPSTEIN
276
(4)
J(x,, 4)
max ZA(x2,
1
-
x2, z) =
v(X2*, 1 -x2*,
zr),
X2<Xl
where A solves the problem in (4). Then x*<x,
1 -x2, zi)/dX2)x2=x, <0. By the eilvelope theorem
0,
(5)
(6)
Jix(x1, 4) =
J,(x*, )
if and only if 4i((dv(x2,
or evaluating at x1 = x,
< XI,
X2*
E
Qi(dvldX
=
max(0, >3i#(dv/dx2)x2=
2.Y2
= X
lX
2
x*=
X1
x)
The second function inside the maximum is linear and hence convex in 4. Since
a maximumnof two convex functions is itself convex, it follows that Jx, is convex
in 4. Therefore, by Theorem 1, x, is larger the more informative is Y.
The piecewise linearity of Jxl is also consistent with intuition. For example,
suppose that Z +?2(columns
of H) are suclh that x*=x* given either 4=ir,
or X=r2 in (4). Define H' by 7 ' =r' ==(1/2)m1 +(I/2)m2 and 7r'=7ri i = 3,...,
and choose q and q' so that Hq=T'q'(=r).
Y is more informative than Y'
since, roughly speaking, two messages of Y are "garbled" into a single message
in Y'. (See also MM (p. 154).) But from (6) and Theorem 1, x4 is the same
for Y and for Y'. The "garbling" of messages both of whiclh, upon receipt,
would result in a binding constraint in period 2, has no effect on tlhe period I
decision, and similarly for two messages both of which would result in an ineffective constraint in period 2. Only when the process of "garbling" involves
at least one message yi that corresponds to x4*<x and another yj such that
2 =1, is the loss of informationconsequentialfor behavior. Such "garbling"
strictly increases the amount of first period highway construction undertaken.
5.
THE TIMING OF ORDERS FOR CAPITAL
There is another interpretation, which we now describe briefly, tllat may be
provided for the mathematical structure of the problem in Section 4.
A producer may order capital equipment in period 1 or in period 2 at prices
cl and c2 respectively. The equipment is for production in period 3. Labor,
tlheother factor of production, may be adjusted instanteously to the level desired
in period 3. In periods 1 and 2 there is uncertainty about the ouptut and labor
prices that will prevail in period 3. C2>C1, i.e., the earlier the order is placed,
the lower the price that must be paid for the capital. Tlle producer will still, in
general, order some capital in period 2 because by that time he will have revised
his expectations about period 3 prices.
Formally, the producer solves
(7)
max - c1x1 + Y qj max {Y_ij9g(zi; x1 +
X2)
-
C2X22}
xi denotes the amount ordered in period i, i= 1, 2. The period 1 order is irreversible(x2 ?0). g(p, w; x) is the variable profit function corresponding to the
277
TEMPORAL RESOLUTION OF UNCERTAINTY
producer'stechnology; it gives the maximum variable profits attainable in period 3
given capital stock x and outpLut anid labor pr-ices p and *v respectively.'0 Z
is a vector random vatriable, with possible vector realizaitions (z,,..., z,,,), representinlguncertainityaibout the period 3 valuLes of these prices. qj anid 7ij
correspond, as before, to a r.v. Y whose realizaitiotnbecomes knownl before the
period 2 order must be placed.
+x2, problem (7) hCasa
With the chanlge of decision variables u, =x,,I2=x,
structure similar to that of (3) with the exception that a2?a, whereas x2<x, in
4) is a minimLum of two
(3). Define J(cx,, 4) in the obvioLusfashion. Then Jofl(oc*,
linear functions of 4 and henlceis concave in . Tlherefore,the more informative
is Y the smaller is the optimazlfirst period order, i.e., the greater is the flexibility
left for the second pericod.
6.
A CONSUMPTION-SAVINGS
PROBLEM
An individuLalhas a given wealtlhw, which lhewishes to allocate between consuLmptionin three periods. What hiedoes not consulme in any period he invests
in a single asset. Investrnent in period I yields a suLregross retuLrnr- and inZ."1 The allocationlis nmadeso
vestment in period 2 yields a random gross retuLrn
as to maximize expected ultility, i.e., the consulmersolves
(8)
max (wv,
O:~X<1x
fsl
-
max 110.XI --x2) + 13Y_-qjO<x2 <r.r
X
x,) + ---
- ZTrijtI(X2Zi)}-
i
x1 and x2 denote savings in periods I and 2 respectively, r is the sure gross return
to first period savings, zi, i= I,..., in are the possible gross retuLrnsto second
period savings and fi is a ultilitydiscount factor. The ultility index has constant
relative risk aversion RRA =-vu"(wv)/u'(iv), i.e.,
it(IV) =
Iv'-a/(0-),
a # 1,
log,I=t
and RRA= a>0. Again, somneinformation abotutZ is gained, thloulghl observilng
Y before x2 is chosen.
One might expect x1 to increase as Y becomes more informative sinicethere is
an incentive to carry over more wealtlhto period 2 to take advantage of tlle increased iniformation. On the other hand, the inlcreasein information and the
resulting rise in the expected valuLeof the suimof second and third period ultility
1l Sce Diewert[1974]. If F(x, L) is thc productionfunction, g(p, ,,; x) max {1)F(x, L)
- iiLjL>0}.
g uniquelydefinesthe tcchnologyand is characterizedby the following propeities:
linear homogeneousand convex in prices,concave in x, increasingin p, x and decro.asingin )ii.
g is strictlyconcave in x if F is strictlyconcave.
11 The assumptionthat r is certainis madefor simplifyingpurposesonly. If the rateof return
about Z. All Ourresults, suitably
were a r.v. R it would in generalprovidesome informllationi
Y and Y' such that Y is nmorcinformative
modified,remain valid if we compare experimilents
than Y' conditionalon R.
LARRY G. EPSTEIN
278
for any giveni xl imply that the consumer can worry less about the future and
indulge in more consumption in period 1. Thus there is also an incentive to
reducex
12
A refereehas provided the following furtherintuition: the above effects of better
information may be interpreted as suLbstitutionand income effects respectively.
This suggests that the elasticity of substitution will help to determine the final
result. The three-period uttilityindex defined above is ordinally equivalent to a
CES uttility funlctioin with elasticity ot substitLutionequLalto (I la). Intuitioll
suggests, therefore, that the substitutioll (income) effect will dominate if a is small
(large). This we now proceed to demonstrate.13
Define
(9)
J(x I,
i)-nx
|'IK('X
.
O<X2 <rxi
-x2)
+ -/*-
l
iu(X2Zi)7
and denote by x2 the SOlution in (9). Straightforward computatiolns yield the
/
"
if a o 1.
if a = I and x2 = 'x [I + ( (iz
solutionis XA= }x1 /(I ?/)
+
By the enivelopetheorem,
Jx, (xl,
) =ru'(rx,
- xA) =r1~xW
(Zr
I, ?
Li
j-e)/
for all
o.
JK,(x, ) is thus convex (concave) in 4 if oc<(>)1, and is independent of
if o- 1. By Theorem 1, x4 is not influenced by ftuture information if oc= 1.
(Note that X4 above is independenitof - if a = 1.) Also the prospect of more
information in the future increases (reduces) savings if x<(>)1.
It is interesting to compare the behavioral response to an earlier resolutioll of
uncertainty with the response to increased prior uncertainty about Z. We would
like to vary the riskiness of Z alone, keeping the informativeness of Y constant
in some sense. This partial variation in the probability distributioll would seem
to be accomplished by keeping the likelihood matrix A fixed as r changes in the
sense of Rothschild and Stiglitz [1970]. However, we have not been able to
determine the behavioral impact of such a change. Therefore, we consider only
the polar cases where (a) Y provides no information about Z, and (b) where Y
provides perfect information about Z. In (a) Y may be ignored completely
since it doesn't enter the uitility function and in (b) Z is essentially observed before
period 2.
In both cases (a) and (b) the decision problem (8) reduces essentially to a twoperiod framework. Therefore the now familiar techniques of Rothschild and
Stiglitz [1971] (especially section 2.A) may be applied to determine the effects of
increased first period uLlcertaintyabout Z. In fact, as indicated briefly in the
W
12 The resultantambiguityis akin to that surroundingthe effects on savings of greater uncertaintyin a two-periodmodel. See Sandmo [1970].
13 The refereehas conjecturedthat the results below hold more generallyand depend only
on the elasticity of substitution. It is not obvious, however, that the results depend only on
of the utility function.
ordinal proper-ties
TEMPORAL RESOLUTION OF UNCERTAINTY
279
Appendix, the optimal savings ruLlescan be determinledexplicitly in both (a) and
(b).
We find that reduced (increased) prior uncertaintyin case (a) has the same qualitative impact on behavior as does an earlier (later) resolution of uncertainty. The
same is true in case (b) if o?> 1/2. However, if O<o< < 1/2, the qualitative impact
of a change in the prior uncertainty surrounldingZ is ambiguous.
Two general conclusionis may be drawni: First, the qualitative effects of increased prior untcertaintydepenidon the temnporalrcsoluLtioll Of thaIt unlcertainlty.14
Second, and more directly related to the theme of this paper, the relationship
between the qualitative effects of a change in prior uncertainty on the one
hand and of the timing of the resoluLtionof uncertainty on the other hand, is
ambiguLouLs.
7.
A FIRM'S DEMAND FOR CAPITAL
Consider a modification of the model of Sectionl5. A firm produces a single
output by miieansof the strictly concave productioll FulnctioniF(K, L). Output
is produced and sold in period 3 at a price P whiclhbecomes known to the film
only in period 3. Both factors of production must be hired or purchased in
advance of the actual production date but the lag is larger for capital. Therefore,
capital K must be ordered in period I and labor L in period 2. By the time
the labor decision must be made the producer will have revised his prior expectations about the price as a result of having observed a correlated r.v. Y. The firm
maximizes expected profits.
The formal problem solved by the firm is the following:
?
max - cK +
max [Y mijPiF(K,L) - vL] ,
(10)
qj nO
0O<K
j
O?L
i
where *v denotes the wage rate and c the rental or purchase price of capital,
which, along with the possible output prices pi, are assumed to be appropriately
discounted.
Define
(I I)
J(K, *v,4)
max YipjF(K, L) - vL.
L20
i
From the definition of the variable profit function g(p, *v;K) (see footnote 10) it
is clear that J(K, *v,4)=g(y4ipj, *v;K). Thlerefore,JK is convex (concave) in
4 if gK(p, *v;K) is convex (concave) in p. In period (1), the firmnsolves
max EqjJ(K, wv,ij)
-
cK.
We conclude therefore from Theorem I that as Y becomes more informative,
14 For the two polar cases of temporal resolution, this fact has been recognized implicitly,
if not explicitly, in many analyses of firm behavior under uncertainty. See, for example,
Turnovsky[1973],Hartman[1976]and Epstein[1978].
LARRY G. EPSTEIN
280
K* increases (decreases) if gK(p, V;K) is convex (concave) in the output price p.
The noted third order property of the variable profit functioni has been found
to be important also in the two-period models of the firm analyzed by Hartman
[1976] and Epstein [1978]. Insight into our result and into its relationship with
these two analyses is most easily provided by addressing the corresponding
question to that considered at the end of the last section; namely, the relationship
between the effects of increased prior uncertainty and ani earlier resolution of
uncertainty.
Define cases (a) and (b) as in the previous section. In (b), P is observed at the
start of period 2 and problem (1) becomes
(12)
max _i-g(pi, w; K)
K?O
-
cK,
precisely the two-period problem considered by Hartman and Epstein. Fiom
their results we see that increased prior uncertaintyincreases (reduces) the demand
for capital if gK(p, w; K) is convex (concave) in p. Combining this and our earlier
finding, we may conclude that when Yprovides perfect information about P, later
resolution and a redtiction in prior uncertainty have similar qualitative impacts
on K*.15
This result has a simple explanation: Since all uncertainty is removed before
the labor decision is made, a redtuctionin prior uncertainty means that the capital
decision alone is made stubjectto less uincertainty. A later resolution of uncertainty means that in an average sense, less uincertaintywill be resolved before
L* is chosen. Therefore, again the environment at the time of the capital decision
is made relatively less uncertain than at the time of the labor decision. In both
cases there is an incentive, to the extent that capital and labor are highly substitutable, to employ relatively more capital. In fact if F has constant elasticity of
substitution a and degree of homogeneity ft>1, then it follows from Hartman
[1976] that K* rises in response to a later resolution if v> 1/(1 -i) and that K*
falls if a< 1/(1 -It).
(The appeal of the explanation is diminiishedby the fact that it does not apply
to the model of Section 6. The particular mathematical feature of the production model seems to be that E[P/Y] is a sufficient statistic for the period 2
decision. Onie can show that if Y is more informative than Y', then E[P/Y] is
more variable than E[P/Y'] in the sense of Rothschild and Stiglitz [1970].)
Turn nlow to case (a) where Yprovides no information about P. Then K and
L are essentially chosen simultaneously to solve
(13)
max (EP)F(K, L)
-
wL - cK,
L>O
K2O
where EP= Yripi is the expected valtue of the prior price expectationi. Prior
15 Note that we are comparingthe effects of a change in prior variabilitygiven perfect information in period 2, with the effects of a change in the structuLeof infornmationgiven any
initial resolutionof uncertainty.
TEMPORAL RESOLUTION OF UNCERTAINTY
281
uncertainty has no effect on behavior as the expected value of the output price
is a sufficient statistic for input decisions.
These findings confirm our earlier concltusions stated at the end of Section 6.
In addition they demonstrate the following: the relationship between the qualitative effects of earlier resolution and reduced prior uncertainty fot' a giveni
information structure over- timiie(e.g., perfect informationi before period 2) is
model specific.
Inistitutefor PolicY Analysis, Univer.sity of Tor-onto,Caniacda
APPENDIX
Suppose that oc l. Consider the decision problem (8) in case (a); i.e., when
Y provides no informationi about Z. Define V(x2)_rax,1 u(w1-x1)?u(rx1
-o), where b is a
-x2)/f5. Then onie can show that V(x2)=b(w1r-x2)1-"/(1
constant (independent of x2), and that the optimal x1 is given by xt(x2)=
The consumer's choice problenm may be ex+r].
+ x2]1[(r/)/
[(r/l)wI
pressed in the form
max b(w1lr- x2)_+2
1 x
X2
fi2
E(X2Z)_
-a
a two-period consumption-savings with constant relative risk aversion ac utility
funictioni. From Rothschild and Stiglitz [1971, pp. 68-72], x4 increases (falls)
Since X4(X2) is increasing in x2,
as Z becomes more uLncertainif i>(<)1.
optimal first period saving responds sinmilarlyto chaingesin prior uncertainltyabout
Z.
In case (b), the consumer solves
max u(w1 - x1) +
(14)
XI
/
E V(rx1, Z),
One can show that V(w, z)=
where V(wv,z)_maxx2 u(wv-x2)+(1/fi)u(zx2).
A(z)w'-0/(t -4), A(z)= [zf l-a)/ +I/SP1]0. Thus (14) becomes
max
-____
+
xl-EA(Z)
It follows that x4 varies positively witlhEA(Z). A(z) is convex if LX>1, concave
if 1/2<Lx< 1, aind neither if 0<<L < 1/2. The stated (Section 6) influence on 4x
of uncertainty about Z follows.
The analyses are similar for u= 1.
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